4d $\mathcal{N}$=2 theories with disconnected gauge groups

TL;DR

The paper develops a comprehensive framework for gauging discrete subgroups of global symmetries in 4d rank-1 ${\cal N}=2$ SCFTs, showing that only special combinations of ${U(1)_R}$, EM-duality ${SL(2,\mathbb{Z})}$, and outer automorphisms of flavor symmetry can be consistently gauged. By enforcing CB-geometry and RG-flow consistency, the authors construct numerous new theories, including ones with exceptional flavor groups ${F_4}$ and ${G_2}$ and several ${\cal N}=3$ fixed points, organized into ${I_0^*}$, ${I_2^*}$, ${IV^*}$, ${III^*}$, and ${II^*}$ deformation-series. They demonstrate that discretely gauged theories retain local dynamics but modify the CB scaling and operator spectrum, leading to a modification of the Shapere-Tachikawa relation and proposing a CB-twisted partition function-based central-charge formula for the gauged cases. The work also provides detailed Higgs branch analyses for both lagrangian and non-lagrangian cases, including explicit examples like ${[I_0^*,D_4]}\to [III^*,B_3]$ and ${[II^*,F_4]}$, and discusses central charges and RG flows, offering a rich expansion of the rank-1 SCFT landscape with practical implications for duality and symmetry-breaking structures.

Abstract

In this paper we present a beautifully consistent web of evidence for the existence of interacting 4d rank-1 $\mathcal{N}=2$ SCFTs obtained from gauging discrete subgroups of global symmetries of other existing 4d rank-1 $\mathcal{N}=2$ SCFTs. The global symmetries that can be gauged involve a non-trivial combination of discrete subgroups of the $U(1)_R$, low-energy EM duality group $SL(2,\mathbb{Z})$, and the outer automorphism group of the flavor symmetry algebra, Out($F$). The theories that we construct are remarkable in many ways: (i) two of them have exceptional $F_4$ and $G_2$ flavor groups; (ii) they substantially complete the picture of the landscape of rank-1 $\mathcal{N}=2$ SCFTs as they realize all but one of the remaining consistent rank-1 Seiberg-Witten geometries that we previously constructed but were not associated to known SCFTs; and (iii) some of them have enlarged $\mathcal{N}=3$ SUSY, and have not been previously constructed. They are also examples of SCFTs which violate the Shapere-Tachikawa relation between the conformal central charges and the scaling dimension of the Coulomb branch vev. We propose a modification of the formulas computing these central charges from the topologically twisted Coulomb branch partition function which correctly compute them for discretely gauged theories.

Figures (3)

• Figure 1: Singularities on a deformed $I_6$ CB for two different mass deformations. The red circles mark the positions of the $I_2$ singularities, and the black dot marks the origin.
• Figure 2: Green, blue and red arrow label matching, compatible and unphysical RG flows while green and blue backgrounds indicate good and ugly theories respectively. While there is always a matching RG flow pattern for all good theories in the figure, there are other flows which are necessarily only compatible for the ugly ones.
• Figure 3: The figures on the left are CB geometries of the $I_4$-series and $I_2$-series deformed $I_0^*$ geometry for special values of ${\tau}$, and their daughter geometries are on the right. The red circles mark the positions of the singularities, and the black dot marks the origin.